Properties

Label 4-504e2-1.1-c5e2-0-12
Degree 44
Conductor 254016254016
Sign 11
Analytic cond. 6534.046534.04
Root an. cond. 8.990748.99074
Motivic weight 55
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 10·5-s + 98·7-s − 54·11-s + 240·13-s − 1.90e3·17-s − 400·19-s − 2.93e3·23-s − 1.89e3·25-s − 5.54e3·29-s + 4.90e3·31-s + 980·35-s + 2.95e3·37-s − 6.07e3·41-s + 3.30e3·43-s − 1.69e4·47-s + 7.20e3·49-s − 6.89e3·53-s − 540·55-s − 5.38e4·59-s + 4.14e3·61-s + 2.40e3·65-s + 3.35e4·67-s − 7.35e4·71-s + 128·73-s − 5.29e3·77-s + 5.77e4·79-s − 2.30e4·83-s + ⋯
L(s)  = 1  + 0.178·5-s + 0.755·7-s − 0.134·11-s + 0.393·13-s − 1.59·17-s − 0.254·19-s − 1.15·23-s − 0.606·25-s − 1.22·29-s + 0.916·31-s + 0.135·35-s + 0.354·37-s − 0.563·41-s + 0.272·43-s − 1.12·47-s + 3/7·49-s − 0.337·53-s − 0.0240·55-s − 2.01·59-s + 0.142·61-s + 0.0704·65-s + 0.912·67-s − 1.73·71-s + 0.00281·73-s − 0.101·77-s + 1.04·79-s − 0.366·83-s + ⋯

Functional equation

Λ(s)=(254016s/2ΓC(s)2L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(254016s/2ΓC(s+5/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 254016254016    =    2634722^{6} \cdot 3^{4} \cdot 7^{2}
Sign: 11
Analytic conductor: 6534.046534.04
Root analytic conductor: 8.990748.99074
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 254016, ( :5/2,5/2), 1)(4,\ 254016,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3 1 1
7C1C_1 (1p2T)2 ( 1 - p^{2} T )^{2}
good5D4D_{4} 12pT+1994T22p6T3+p10T4 1 - 2 p T + 1994 T^{2} - 2 p^{6} T^{3} + p^{10} T^{4}
11D4D_{4} 1+54T+284302T2+54p5T3+p10T4 1 + 54 T + 284302 T^{2} + 54 p^{5} T^{3} + p^{10} T^{4}
13D4D_{4} 1240T+739862T2240p5T3+p10T4 1 - 240 T + 739862 T^{2} - 240 p^{5} T^{3} + p^{10} T^{4}
17D4D_{4} 1+1906T+1860002T2+1906p5T3+p10T4 1 + 1906 T + 1860002 T^{2} + 1906 p^{5} T^{3} + p^{10} T^{4}
19D4D_{4} 1+400T+3279798T2+400p5T3+p10T4 1 + 400 T + 3279798 T^{2} + 400 p^{5} T^{3} + p^{10} T^{4}
23D4D_{4} 1+2930T+9774686T2+2930p5T3+p10T4 1 + 2930 T + 9774686 T^{2} + 2930 p^{5} T^{3} + p^{10} T^{4}
29D4D_{4} 1+5540T+40407182T2+5540p5T3+p10T4 1 + 5540 T + 40407182 T^{2} + 5540 p^{5} T^{3} + p^{10} T^{4}
31D4D_{4} 14904T+59914302T24904p5T3+p10T4 1 - 4904 T + 59914302 T^{2} - 4904 p^{5} T^{3} + p^{10} T^{4}
37D4D_{4} 12952T+138794486T22952p5T3+p10T4 1 - 2952 T + 138794486 T^{2} - 2952 p^{5} T^{3} + p^{10} T^{4}
41D4D_{4} 1+6070T+232254602T2+6070p5T3+p10T4 1 + 6070 T + 232254602 T^{2} + 6070 p^{5} T^{3} + p^{10} T^{4}
43D4D_{4} 13304T+197837766T23304p5T3+p10T4 1 - 3304 T + 197837766 T^{2} - 3304 p^{5} T^{3} + p^{10} T^{4}
47D4D_{4} 1+16988T+429309854T2+16988p5T3+p10T4 1 + 16988 T + 429309854 T^{2} + 16988 p^{5} T^{3} + p^{10} T^{4}
53D4D_{4} 1+6896T+613869254T2+6896p5T3+p10T4 1 + 6896 T + 613869254 T^{2} + 6896 p^{5} T^{3} + p^{10} T^{4}
59D4D_{4} 1+53820T+1699029142T2+53820p5T3+p10T4 1 + 53820 T + 1699029142 T^{2} + 53820 p^{5} T^{3} + p^{10} T^{4}
61D4D_{4} 168pT+1287381294T268p6T3+p10T4 1 - 68 p T + 1287381294 T^{2} - 68 p^{6} T^{3} + p^{10} T^{4}
67D4D_{4} 133516T+2261444678T233516p5T3+p10T4 1 - 33516 T + 2261444678 T^{2} - 33516 p^{5} T^{3} + p^{10} T^{4}
71D4D_{4} 1+73518T+4934300734T2+73518p5T3+p10T4 1 + 73518 T + 4934300734 T^{2} + 73518 p^{5} T^{3} + p^{10} T^{4}
73D4D_{4} 1128T360358674T2128p5T3+p10T4 1 - 128 T - 360358674 T^{2} - 128 p^{5} T^{3} + p^{10} T^{4}
79D4D_{4} 157740T+6120687198T257740p5T3+p10T4 1 - 57740 T + 6120687198 T^{2} - 57740 p^{5} T^{3} + p^{10} T^{4}
83D4D_{4} 1+23016T4303421546T2+23016p5T3+p10T4 1 + 23016 T - 4303421546 T^{2} + 23016 p^{5} T^{3} + p^{10} T^{4}
89D4D_{4} 1141530T+14809201898T2141530p5T3+p10T4 1 - 141530 T + 14809201898 T^{2} - 141530 p^{5} T^{3} + p^{10} T^{4}
97D4D_{4} 1+226216T+29691907182T2+226216p5T3+p10T4 1 + 226216 T + 29691907182 T^{2} + 226216 p^{5} T^{3} + p^{10} T^{4}
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   L(s)=p j=14(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.698621357691645855154396779744, −9.632426649691190045388261930508, −8.924595907220051428186947398244, −8.605047807448879723279010070588, −7.999466963441445661836865760849, −7.85000536979697586208624090541, −7.19640931536114569642355592199, −6.61340208993397008404142752301, −6.13989783238775839149158436717, −5.88644039374264784313795615549, −4.95751104261027156037606843425, −4.84544816534123082318780079736, −3.97798755369182886633256897094, −3.85924955071786598137556514679, −2.83323491538201949059364176099, −2.30328644003832381056158689407, −1.72523941582283573050028917718, −1.26019755765368447561610254117, 0, 0, 1.26019755765368447561610254117, 1.72523941582283573050028917718, 2.30328644003832381056158689407, 2.83323491538201949059364176099, 3.85924955071786598137556514679, 3.97798755369182886633256897094, 4.84544816534123082318780079736, 4.95751104261027156037606843425, 5.88644039374264784313795615549, 6.13989783238775839149158436717, 6.61340208993397008404142752301, 7.19640931536114569642355592199, 7.85000536979697586208624090541, 7.999466963441445661836865760849, 8.605047807448879723279010070588, 8.924595907220051428186947398244, 9.632426649691190045388261930508, 9.698621357691645855154396779744

Graph of the ZZ-function along the critical line